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Math Help - Topology question

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    Topology question

    Prove that Cl(Q)=R in the standard topology on R.
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    Quote Originally Posted by horowitz View Post
    Prove that Cl(Q)=R in the standard topology on R.
    Let a\in R for any any \epsilon > 0 we know that (a-\epsilon,a+\epsilon) contains both rational and irrational points. Therefore, all points are boundary points. Thus, \partial \mathbb{Q} = \mathbb{R} which means \text{cl}( \mathbb{Q}) = \mathbb{Q} \cup \partial \mathbb{Q} = \mathbb{R}.
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